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5 5 duoprism

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5-5 duoprism

In geometry of 4 dimensions, a 5-5 duoprism is a polygonal duoprism, a 4-polytope resulting from the Cartesian product of two pentagons.

Contents

It has 25 vertices, 50 edges, 35 faces (25 squares, and 10 pentagons), in 10 pentagonal prism cells. It has Coxeter diagram , and symmetry [[5,2,5]], order 200.

Images

Seen in a skew 2D orthogonal projection, 20 of the vertices are in two decagonal rings, while 5 project into the center. The 5-5 duoprism here has an identical 2D projective appearance to the 3D rhombic triacontahedron. In this projection, the square faces project into wide and narrow rhombi seen in penrose tiling.

The regular complex polytope 5{4}2, , in C 2 has a real representation as a 5-5 duoprism in 4-dimensional space. 5{4}2 has 25 vertices, and 10 5-edges. Its symmetry is 5[4]2, order 50. It also has a lower symmetry construction, , or 5{}×5{}, with symmetry 5[2]5, order 25. This is the symmetry if the red and blue 5-edges are considered distinct.

The birectified order-5 120-cell, , constructed by all rectified 600-cells, a 5-5 duoprism vertex figure.

5-5 duopyramid

The dual of a 5-5 duoprism is called a 5-5 duopyramid. It has 25 tetragonal disphenoid cells, 50 triangular faces, 35 edges, and 10 vertices.

It can be seen in orthogonal projection as a regular 10-gon circle of vertices, divided into two pentagons, seen with colored vertices and edges:

The regular complex polygon 2{4}5 has 10 vertices in C 2 with a real represention in R 4 matching the same vertex arrangement of the 5-5 duopyramid. It has 25 2-edges corresponding to the connecting edges of the 5-5 duopyramid, while the 10 edges connecting the two pentagons are not included. The vertices and edges makes a complete bipartite graph with each vertex from one pentagon is connected to every vertex on the other.

References

5-5 duoprism Wikipedia
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